Component and System Reliability Assessment of Landfill Cover Systems Using Pseudostatic and Pseudodynamic Methods

Data Availability Statement

Acknowledgments

Notation

A_Act, A_Pass

area of active and passive wedges (m²);

a_h(y, t), a_v(y, t)

horizontal and vertical seismic acceleration at any depth, y, and time, t (m/s²);

B_f, B_af

failure and mass above failure blocks;

c

cohesion of veneer cover soil (kPa);

c₀

interface cohesion between the drainage layer and the overlying cover soil (kPa);

E_A, E_P

interwedge forces acting on active and passive wedges (kN/m);

F_JR, F_PJ

normal forces acting on the sliding planes JR and PJ (kN/m);

$F_{IJ},F_{IJ}^{\mathrm{\prime }},F_{P{P}^{\mathrm{\prime }}}{F}_{P^{\mathrm{\prime }}J}$

normal forces acting on the faces IJ, PP', and P'J of failure block (kN/m);

FS_ds

factor of safety against DSF (dimensionless);

FS_uf

factor of safety against UFF (dimensionless);

f_a

soil amplification factor (dimensionless);

G

shear modulus of cover soil (dimensionless);

g

acceleration due to gravity (m/s²);

g(x)

performance function;

g_ds(x)

performance function for DSF;

g_uf(x)

performance function for UFF;

H

height of landfill (m);

h

thickness of cover soil (m);

h_w

height of the seepage level measured from the bottom of the drainage layer (m);

I_r

immersion ratio (dimensionless);

k_h, k_v

horizontal and vertical seismic acceleration coefficients (dimensionless);

l_JR, l_PJ

lengths of the faces JR and PJ of sliding plane PJR (m);

$l_{PI}\mathrm{and}{l}_{P^{\mathrm{\prime }}J}$

length of uplifted block (m);

L

length of veneer cover slope (m);

L_g

Lagrange multiplier;

NS

number of simulations;

$P_{f\mathrm{\_}ds},P_{f\mathrm{\_}uf}$

probability of failure for DSF and UFF;

$P_{f\mathrm{\_}ds\mathrm{\_}\mathrm{FORM}},P_{f\mathrm{\_}uf\mathrm{\_}\mathrm{FORM}}$

probability of failure for DSF and UFF computed using FORM;

$P_{f\mathrm{\_}ds\mathrm{\_}\mathrm{MCS}},P_{f\mathrm{\_}uf\mathrm{\_}\mathrm{MCS}}$

probability of failure for DSF and UFF computed using MCS;

$P_{f\mathrm{\_}ds\mathrm{\_}\mathrm{SORM}},P_{f\mathrm{\_}uf\mathrm{\_}\mathrm{SORM}}$

probability of failure for DSF and UFF computed using SORM;

P_fij

joint probability of failure between the failure modes;

$P_{f\mathrm{\_}lb},P_{f\mathrm{\_}ub}$

lower and upper bounds of system probability of failure for DSF and UFF modes;

$P_{f\mathrm{\_}\mathrm{MCS}}$

probability of failure computed using the MCS approach;

$P_{f\mathrm{\_}\mathrm{series}}$

series system probability of failure;

P(E_i)

component probabilities of failure;

PD_ds, PD_uf

percentage differences between the probabilities of failure, computed with and without considering the correlation between the random variables for DSF and UFF;

PD_ds1, PD_uf1

percentage difference between the probabilities of failure computed using FORM and MCS methods for DSF and UFF modes;

PD_ds2, PD_uf2

percentage difference between the probabilities of failure computed using SORM and MCS methods for DSF and UFF modes;

r_cϕ

Correlation coefficient between “c” and “ϕ” of cover soil;

S_JR, S_PJ

shear resisting forces against sliding failure acting on the sliding planes JR and PJ (kN/m);

$S_{IJ},S_{IJ}^{\mathrm{\prime }},S_{P{P}^{\mathrm{\prime }}}$

shear resisting forces acting on the faces IJ and PP′ of the failure block (kN/m);

T_b

basis matrix;

T_r

transformation matrix;

t, T

time and period of lateral shaking (s);

$U_{JR}^x,U_{PJ}^x$

hydrostatic forces acting on the sliding planes JR and PJ for the case number x (kN/m);

$U_{JRdyn}^x,U_{PJdyn}^x$

hydrodynamic forces acting on the sliding planes JR and PJ for the case number x (kN/m);

$U_{IJ}^z,U_{IJ}^{\mathrm{\prime }z},U_{P{P}^{\mathrm{\prime }}}^z,U_{P{P}^{\mathrm{\prime }}}^z$

hydrostatic forces acting on the faces IJ, PP', and P'J of the failure block for the case number z (kN/m);

$U_{IJdyn}^z,U_{IJdyn}^{\mathrm{\prime }z},U_{P{P}^{\mathrm{\prime }}dyn}^z,U_{P^{\mathrm{\prime }}Jdyn}^z$

hydrodynamic forces acting on the faces IJ, PP’, and P'J of the failure block for the case number z (kN/m);

u_i

reduced variables;

V_p, V_s

primary and shear-wave velocities propagating through landfill final cover (m/s);

W_Act, W_Pass

weight of active and passive wedges (kN/m);

W_f, W_af

weight of failure and mass above the failure blocks (kN/m);

X

vector of original random variables;

α_ds, α_uf

directional cosines of DSF and UFF modes;

β_ds, β_uf

reliability indices against DSF and UFF modes;

β_ds,uf, β_uf,ds

joint reliability indices of DSF and UFF modes;

β_FORM, β_SORM

First-order and second-order reliability indicies;

${\beta }_{ds\mathrm{\_}\mathrm{FORM}},{\beta }_{uf\mathrm{\_}\mathrm{FORM}}$

reliability indices against DSF and UFF computed using FORM;

${\beta }_{ds\mathrm{\_}\mathrm{SORM}},{\beta }_{uf\mathrm{\_}\mathrm{SORM}}$

reliability indices against DSF and UFF computed using SORM;

β_HL

Hasofer–Lind reliability index;

β_lb, β_ub

lower and upper bound system reliability indices;

β_MCS

reliability index computed using the MCS approach;

β_series

series system reliability index;

γ

bulk density of soil (kN/m³);

γ_d

dry density of soil (kN/m³);

γ_sat

saturated unit weight of cover soil (kN/m³);

γ_w

unit weight of water (kN/m³);

δ

interface friction angle between the drainage layer and the overlying cover soil (°);

${\epsilon }_{\mathrm{CDF}}$

CDF error;

${\epsilon }_{P-P}$

percentile–percentile error;

${\epsilon }_{Q-Q}$

quantile–quantile error;

η, λ

wavelengths of vertically propagating primary and shear waves through landfill final cover (m);

θ

slope angle of veneer cover (°);

κ_i

principal curvatures of LSF;

μ_i

mean of random variables;

ρ_c

density of cover soil (kg/m³);

ρ_ds,uf

correlation coefficient between DSF and UFF modes;

ρ_ij

correlation coefficient matrix;

ρ_s

density of soil (kg/m³);

ρ_sat

saturated density of cover soil (kg/m³);

ρ_w

density of water (kg/m³);

σ_i

standard deviation of random variables;

Φ

standard normal distribution function;

ϕ

angle of internal friction of veneer cover soil (°);

ψ

angle of sliding plane “PJ” (°); and

ω

angular frequency of lateral shaking (rad/s).

References